Analytic Theory of LTV Systems

Abstract

We recall here some known facts about the stability of LTV systems given by their state representation

$$ \dot{x}(t)=A(t)x(t)+B(t)u(t) $$

$$ y(t)=C(t)x(t)+D(t)u(t) $$

where A(t), B(t), C(t) and D(t) are matrices which entries are real-valued piecewise continuous functions of t and the dimension of the state x is n. The structural properties of such an LTV system are analyzed using the solutions of the system of differential equations (12.1) in Section 12.2. The Lyapunov analysis of stability is recalled in Section 12.4. Definitions of the poles and zeros are given in Section 12.5 using Lyapunov transformations of the initial system. This analytic approach is a complement of the intrinsic algebraic one presented in Chapter 6.